Gene dynamics of haplodiploidy favor eusociality in the Hymenoptera

Abstract The problem of whether haplodiploidy is responsible for the frequent evolution of eusociality in the Hymenoptera remains unresolved. The little‐known “protected invasion hypothesis” posits that because a male will transmit a new allele for alloparental care to all his daughters under haplodiploidy, such an allele has a higher probability of spreading to fixation under haplodiploidy than under diploidy. This mechanism is investigated using the mating system and lifecycles ancestral to eusocial lineages. It is shown that although haplodiploidy increases the probability of fixation of a new allele, the effect is cancelled by a higher probability of the allele arising in a diploid population. However, the same effect of male haploidy results in a 30% lower threshold amount of reproductive help by a worker necessary to favor eusociality if the sex ratio of dispersing first‐brood offspring remains even. This occurs because when first‐brood daughters become workers, the sex ratio of dispersing first‐brood offspring becomes male‐biased, selecting for an overall female‐biased first‐brood sex ratio. Through this mechanism, haplodiploidy may favor eusociality in the absence of a female‐biased sex ratio in dispersing reproductive offspring. The gene‐centric approach used here reveals the critical role of male haploidy in structuring the social group.


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First-brood sex ratio in fitness calculations 2 Because some first-brood females may act as helpers, depending on their genotype, the sex 3 ratio of first-brood dispersing offspring would be biased toward males if the brood adult sex 4 ratio is even. In the extreme case, if all first-brood females acted as helpers, then all 5 dispersing offspring would be male. Assuming equal costs of producing sons and daughters, 6 it is further assumed that the population sex ratio of dispersing first-brood offspring evolves 7 to remain even (West 2009). That is, on average females produce first-brood sex ratios that 8 result in an even sex ratio in dispersing offspring. This is implemented most easily by 9 assuming that each female produces a first-brood sex ratio that results in an even sex ratio 10 of dispersing offspring. This sex ratio, in terms of the proportion of offspring that are 11 female, is = In the present context, fitness is a property of mated pairs of individuals, rather than 17 genotypes, because with fertility selection, although mating is random, gamete fusions are 18 not. With a univoltine lifecycle, fitness consists of two components: the number of offspring 19 produced in the first brood that disperse and the number produced in the second brood, all 20 of which disperse. The first-brood fitness component is − , where is the number of 21 first-brood offspring produced by the foundress. The second-brood fitness component is the 22 number offspring produced independently by the foundress plus additional offspring 23 produced with help from first-brood daughters that do not disperse: + , where is 24 the additional number of offspring produced with help from each first-brood daughter. 25 Because the sex ratio of dispersing offspring is even in every case, the different reproductive 26 values of males and females due to haplodiploidy cancel out when calculating relative 27 fitness and may be ignored. That is, each mated pair, consisting of a male and female, 28 always produces equal numbers of male and female offspring that disperse and mate 29 randomly with individuals from the population to form new mated pairs. Therefore, the 30 fitness of a mated pair with genotypes and is: Note that &' is a function of the genotypes of a mated pair, as is &' because it is a function 35 of &' . The right-hand side of the equation is divided by 2 because each dispersing offspring 36 mates randomly in the population to produce a new mated pair. Mean fitness is the sum of 37 the products of mated-pair fitness and the frequency of the mated pair, assuming random 38 mating, over all possible mated pairs: 39 40 where & and ' are the genotype frequencies of females and males, respectively (with the 3 value of the subscript indicating the number of altruism allele copies carried by a diploid 4 female or a haploid male). However, because the genotypes of dispersing first-brood 5 females are biased against those carrying the altruism allele, fitnesses of mated pairs are 6 not sufficient to determine allele dynamics. Genotype frequencies must be tracked to 7 determine the fate of the altruism allele. 8

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Deterministic recursion equations were used to track genotype frequencies over 10 generations. The recursion equations for haplodiploidy are described in detail here; the 11 recursion equations for diploidy differ only in that diploid male genotypes must be indexed. 12 Recursion equations are derived separately for females and males because of the genotype 13 bias in dispersing first-brood females and because of the ploidy difference with 14 haplodiploidy. To calculate the frequency of genotype in females in the next spring 15 generation, we must know the proportion of genotype in first-brood daughters that 16 disperse, *(!),&' .
, which is a function of genotypes and of a mated pair. The proportion of 17 genotype in second-brood daughters is given by *("),&' .
(this proportion does not reflect 18 any genotype bias since all second-brood offspring disperse). The overall proportion of 19 females with genotype in dispersing females in both broods is the sum of the proportions 20 from each brood weighted by the proportion of fitness contributed by the brood of origin. 21 The proportion of fitness contributed by the first brood is given by &'(!) . Therefore, the 22 overall proportion of females with genotype in dispersing offspring of parents with 23 genotypes and is 24 25 Thus, the frequency of genotype in females in the next spring generation, after random 28 mating in the current spring generation (formation of mated pairs), is 29 30 Since sexes are considered individually and sex ratios are even, both fitness and mean 33 fitness are halved, which cancels out. 34 For males, the frequency of genotype in the next generation is 35 36 where 0,& 1 is the proportion of males with genotype produced by a female with genotype , 39 which is the same for both broods. Allele frequencies for the entire population, including 40 both males and females, are calculated from the sex-specific genotype frequencies, taking 1 into consideration that females are diploid and males are haploid. 2

Partially bivoltine lifecycle
With a bivoltine lifecycle fitness also consists of two components. The first component is the 5 number of offspring produced by first-brood dispersing offspring, that is, the second 6 generation, all of which disperse. The second component is the number of offspring 7 produced in the second brood of the first generation, all of which also disperse. Dispersing 8 first-brood offspring mate at random in the population and produce offspring, where ! is the number of offspring produced independently (without help) by the 10 foundress in each first-generation brood, and " is the number of offspring produced 11 independently by each dispersing first-brood offspring as part of a mated pair, forming the 12 second generation. These brood sizes may differ ( " < ! ) since first-brood dispersing 13 females may experience higher mortality due to dispersal and mating and must establish 14 new nests. The second-generation brood size is halved because these offspring are the 15 product of outbreeding and are therefore evenly allocated to the descendants of the focal 16 mated pair and mates chosen at random from the population. The second brood produced 17 by the founding mated pair is the same as for a univoltine lifecycle. Therefore, the fitness of 18 a mated pair is:

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To use recursion equations to track genotype frequencies across generations we must 24 determine the genotype proportions of offspring in the second generation. These 25 proportions depend on the proportions of genotypes of dispersing first-brood offspring, 26 which depend on the genotypes of the mated pair. The genotype proportions in the second 27 generation also depend on the population-wide genotype frequencies of the dispersing 28 first-brood's random mates in the population. Second-generation females are produced by 29 both dispersing first-brood females and males. The proportion of genotype in second-30 generation females produced by dispersing first-brood daughters of parents with genotypes 31 and is: 32 33 is the proportion of 37 genotype in dispersing first-brood daughters of parents with genotypes and , and 1 1 is 38 the population-wide frequency of genotype in first-brood males. This latter frequency is 39 calculated as 40 41 where &'(!) is the first-brood fitness of a mated pair with genotypes and , which is the 3 same as the component of fitness due to the first brood with a univoltine lifecycle, 0 (!) is 4 the mean first-brood fitness of mated pairs, & and ' are the genotype frequencies of the 5 mated pair, and 1,& 1 is the proportion of males with genotype produced by a female with 6 genotype . 7 The proportion of genotype in second-generation females produced by dispersing 8 first-brood sons is: 9 10  Weighted Mean 5n/6 + nb/12 n is brood size and b is the number of additional offspring produced with help from a daughter. With a codominant altruism allele, a first-brood daughter carrying one copy has a 0.5 probability of acting as a helper. Weighted Mean nn2/4 + n/2 n is foundress brood size, n2 is the second-generation brood size, and b is the number of additional offspring produced with help from a daughter. Weighted Mean nn2/6 + n/2 + nb/12 n is foundress brood size, n2 is the second-generation brood size, and b is the number of additional offspring produced with help from a daughter. With a codominant altruism allele, a first-brood daughter carrying one copy has a 0.5 probability of acting as a helper. Weighted Mean 11n/18 + 5nb/18 n is brood size, b is the number of additional offspring produced with help from a daughter, f is the firstbrood sex ratio and a is the proportion of first-brood daughters that are altruists. Cross 2: A0A0 x A1 1 st Brood (n; a = 1; f = 1) ♀A0A1 1 n 0 0 0 ♂A0 0 Total 1 n 0 0 0 2 nd Brood (n + nb) ♀A0A1 ½ n/2 + nb/2 n/2 + nb/2 ♂A0 ½ n/2 + nb/2 0 Total 1 n + nb n/2 + nb/2 Grand Total for Cross 2 n/2 + nb/2 Weighted Mean nn2/18 + n/2 + 5nb/18 n is foundress brood size, n2 is the second-generation brood size, b is the number of additional offspring produced with help from a daughter, f is the first-brood sex ratio and a is the proportion of first-brood daughters that are altruists.